In mathematical modeling of cognition, it is important to have well-justified criteria for choosing among differing explanations (i.e., models) of observed data. This project investigates those criteria as well as their instantiation in five model selection methods. Two lines of research will be undertaken. In the first, a thorough investigation of model complexity will be conducted. Comprehensive simulations re intended to determine complexity's contribution to model fit and to model selection. An analytical solution will also be sought with the hope of quantifying model complexity. The second line of work examines the utility of each of the five selection methods in choosing among models in three topic areas in cognitive psychology (information integration, categorization, connectionist modeling), the end goal being to identify their merits and shortcomings. Findings should provide a better understanding of model selection than currently available and serve as a useful guide for researchers comparing the suitability of quantitative models of cognition.